How does graphene demonstrate quantum tunneling?

[dawozel]

Hi as an undergrad Physics student learning quantum mechanics for the first time, I'm having a hard time wrapping my head around quantum tunneling, why does it happen, why can it happen? I was also reading that Graphene exhibits tunneling as well, could someone explain this?

 

[dauto]

Imagine a harmonic wave.
\phi = e^{i(kx-\omega t)}
From de Broglie's relation
p = \hbar k
we see that imaginary momentum (which is classically forbidden) is actually a viable solution for that wave but instead of getting an oscillation out of the exponential we get a exponential decay with the traveled distance. So there is a non-zero possibility of the wave crossing a region of negative kinetic energy (that probability decays exponentially with the thickness of the barrier).

 

[Naty1]

You can think of tunneling, like most quantum phenomena, as statistical [probabilistic] in nature. It stems from that fact that a particle never has a zero probability of existence on the opposite side of an intervening barrier due to Heisenberg uncertainty. Only if a potential barrier approaches infinite height or width does the probability of tunneling approach zero.

Some good insights here:

http://en.wikipedia.org/wiki/Quantum_tunnelling

 

[Naty1]

PS: graphene...

I'm not specifically familiar with graphene but my post above provides a clue...seems like a one atom thick sheet of crystalline carbon would be prone to tunneling...

also the examples in Wicki I linked to might provide clues...

yes, just skimmed and saw this: Tunnel diodes, which I have studied...albeit a long time ago..

...when these are very heavily doped the depletion layer can be thin enough for tunnelling...

 

[Jilang]

Imagine a harmonic wave.
\phi = e^{i(kx-\omega t)}
From de Broglie's relation
p = \hbar k
we see that imaginary momentum (which is classically forbidden) is actually a viable solution for that wave but instead of getting an oscillation out of the exponential we get a exponential decay with the traveled distance. So there is a non-zero possibility of the wave crossing a region of negative kinetic energy (that probability decays exponentially with the thickness of the barrier).

What would be the interpretation of an imaginary momentum. An imaginary mass or an imaginary velocity?

 

[dipole]

What would be the interpretation of an imaginary momentum. An imaginary mass or an imaginary velocity?

Neither, velocity is not a well-defined concept in quantum mechanics, and the momentum is certainly not something that looks like \vec{p} = m\dot{\vec{x}}.

The situation dauto is describing is not a physical one, because the wave function he gives is not normalizable - so trying to interpret what the momentum means in that case is rather pointless. However, if you want to know the relation between momentum and position in quantum mechanics, they are actually Fourier transforms of each-other (hence the uncertainty principle).

 

[Jilang]

Neither, velocity is not a well-defined concept in quantum mechanics, and the momentum is certainly not something that looks like \vec{p} = m\dot{\vec{x}}.

The situation dauto is describing is not a physical one, because the wave function he gives is not normalizable - so trying to interpret what the momentum means in that case is rather pointless. However, if you want to know the relation between momentum and position in quantum mechanics, they are actually Fourier transforms of each-other (hence the uncertainty principle).

So would an imaginary momentum imply an imaginary position too?
And thinking on it would seem that velocity is no less well defined than momentum or position. So please explain what you mean.

 

[dipole]

No. And introducing imaginary momentum is a very poor way of looking at things IMO. When you have a wave function which decays exponentially, as in the case of the finite potential well for a bound particle, then momentum isn't a very meaningful quantity in the classically forbidden region. In fact, if you're going to introduce imaginary momentum, then you have to concede that the momentum operator is no longer hermitian, and it makes one wonder what exactly we're talking about anymore when referencing "momentum".

Have you ever actually studied quantum mechanics? You explain what you mean, when you make the (false) claim that velocity is well defined in the context of quantum mechanics.

Velocity is only something you can only define in a statistical sense. An electron has no such property as velocity. The best you can do is to look at the group velocity of the wave function, which often for an energy-eigenstate will be zero, such as the particle in a box.

 

[Jilang]

Have you ever actually studied quantum mechanics? You explain what you mean, when you make the (false) claim that velocity is well defined in the context of quantum mechanics.

Velocity is only something you can only define in a statistical sense.

I didn't claim otherwise, just that it was no less well defined than position or momentum. I was thinking about it being defined in the QM sense V = i[H,Q].

 

[Jilang]

No. And introducing imaginary momentum is a very poor way of looking at things IMO. When you have a wave function which decays exponentially, as in the case of the finite potential well for a bound particle, then momentum isn't a very meaningful quantity in the classically forbidden region. In fact, if you're going to introduce imaginary momentum, then you have to concede that the momentum operator is no longer hermitian, and it makes one wonder what exactly we're talking about anymore when referencing "momentum".

I am not introducing an interpretation or a way of looking at things here. Mathematically the solution for the momentum is imaginary. The momentum operator is no longer Hermitian but doesn't stop it being unitary so QM still applies in the classically forbidden region. Reading some things about tunnelling today and it looks like the velocity is imaginary and the mass is real. The time spent in the classically forbidden region is imaginary and the width of the gap is real.

 

[DeepCut]

We don't even know the intricacies of lightning so to try and puzzle out the universe will take us a very long time.